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Theorem ipasslem1 24250
Description: Lemma for ipassi 24260. Show the inner product associative law for nonnegative integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
ip1i.1  |-  X  =  ( BaseSet `  U )
ip1i.2  |-  G  =  ( +v `  U
)
ip1i.4  |-  S  =  ( .sOLD `  U )
ip1i.7  |-  P  =  ( .iOLD `  U )
ip1i.9  |-  U  e.  CPreHil
OLD
ipasslem1.b  |-  B  e.  X
Assertion
Ref Expression
ipasslem1  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( ( N S A ) P B )  =  ( N  x.  ( A P B ) ) )

Proof of Theorem ipasslem1
Dummy variables  j 
k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nn0cn 10608 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  k  e.  CC )
2 ax-1cn 9359 . . . . . . . . . . . 12  |-  1  e.  CC
3 ip1i.9 . . . . . . . . . . . . . 14  |-  U  e.  CPreHil
OLD
43phnvi 24235 . . . . . . . . . . . . 13  |-  U  e.  NrmCVec
5 ip1i.1 . . . . . . . . . . . . . 14  |-  X  =  ( BaseSet `  U )
6 ip1i.2 . . . . . . . . . . . . . 14  |-  G  =  ( +v `  U
)
7 ip1i.4 . . . . . . . . . . . . . 14  |-  S  =  ( .sOLD `  U )
85, 6, 7nvdir 24030 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  (
k  e.  CC  /\  1  e.  CC  /\  A  e.  X ) )  -> 
( ( k  +  1 ) S A )  =  ( ( k S A ) G ( 1 S A ) ) )
94, 8mpan 670 . . . . . . . . . . . 12  |-  ( ( k  e.  CC  /\  1  e.  CC  /\  A  e.  X )  ->  (
( k  +  1 ) S A )  =  ( ( k S A ) G ( 1 S A ) ) )
102, 9mp3an2 1302 . . . . . . . . . . 11  |-  ( ( k  e.  CC  /\  A  e.  X )  ->  ( ( k  +  1 ) S A )  =  ( ( k S A ) G ( 1 S A ) ) )
111, 10sylan 471 . . . . . . . . . 10  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( k  +  1 ) S A )  =  ( ( k S A ) G ( 1 S A ) ) )
125, 7nvsid 24026 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
1 S A )  =  A )
134, 12mpan 670 . . . . . . . . . . . 12  |-  ( A  e.  X  ->  (
1 S A )  =  A )
1413adantl 466 . . . . . . . . . . 11  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( 1 S A )  =  A )
1514oveq2d 6126 . . . . . . . . . 10  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( k S A ) G ( 1 S A ) )  =  ( ( k S A ) G A ) )
1611, 15eqtrd 2475 . . . . . . . . 9  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( k  +  1 ) S A )  =  ( ( k S A ) G A ) )
1716oveq1d 6125 . . . . . . . 8  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( ( k  +  1 ) S A ) P B )  =  ( ( ( k S A ) G A ) P B ) )
18 ipasslem1.b . . . . . . . . . . . . 13  |-  B  e.  X
19 ip1i.7 . . . . . . . . . . . . . 14  |-  P  =  ( .iOLD `  U )
205, 19dipcl 24129 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  e.  CC )
214, 18, 20mp3an13 1305 . . . . . . . . . . . 12  |-  ( A  e.  X  ->  ( A P B )  e.  CC )
2221mulid2d 9423 . . . . . . . . . . 11  |-  ( A  e.  X  ->  (
1  x.  ( A P B ) )  =  ( A P B ) )
2322adantl 466 . . . . . . . . . 10  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( 1  x.  ( A P B ) )  =  ( A P B ) )
2423oveq2d 6126 . . . . . . . . 9  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( ( k S A ) P B )  +  ( 1  x.  ( A P B ) ) )  =  ( ( ( k S A ) P B )  +  ( A P B ) ) )
255, 7nvscl 24025 . . . . . . . . . . . 12  |-  ( ( U  e.  NrmCVec  /\  k  e.  CC  /\  A  e.  X )  ->  (
k S A )  e.  X )
264, 25mp3an1 1301 . . . . . . . . . . 11  |-  ( ( k  e.  CC  /\  A  e.  X )  ->  ( k S A )  e.  X )
271, 26sylan 471 . . . . . . . . . 10  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( k S A )  e.  X )
285, 6, 7, 19, 3ipdiri 24249 . . . . . . . . . . 11  |-  ( ( ( k S A )  e.  X  /\  A  e.  X  /\  B  e.  X )  ->  ( ( ( k S A ) G A ) P B )  =  ( ( ( k S A ) P B )  +  ( A P B ) ) )
2918, 28mp3an3 1303 . . . . . . . . . 10  |-  ( ( ( k S A )  e.  X  /\  A  e.  X )  ->  ( ( ( k S A ) G A ) P B )  =  ( ( ( k S A ) P B )  +  ( A P B ) ) )
3027, 29sylancom 667 . . . . . . . . 9  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( ( k S A ) G A ) P B )  =  ( ( ( k S A ) P B )  +  ( A P B ) ) )
3124, 30eqtr4d 2478 . . . . . . . 8  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( ( k S A ) P B )  +  ( 1  x.  ( A P B ) ) )  =  ( ( ( k S A ) G A ) P B ) )
3217, 31eqtr4d 2478 . . . . . . 7  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( ( k  +  1 ) S A ) P B )  =  ( ( ( k S A ) P B )  +  ( 1  x.  ( A P B ) ) ) )
33 oveq1 6117 . . . . . . 7  |-  ( ( ( k S A ) P B )  =  ( k  x.  ( A P B ) )  ->  (
( ( k S A ) P B )  +  ( 1  x.  ( A P B ) ) )  =  ( ( k  x.  ( A P B ) )  +  ( 1  x.  ( A P B ) ) ) )
3432, 33sylan9eq 2495 . . . . . 6  |-  ( ( ( k  e.  NN0  /\  A  e.  X )  /\  ( ( k S A ) P B )  =  ( k  x.  ( A P B ) ) )  ->  ( (
( k  +  1 ) S A ) P B )  =  ( ( k  x.  ( A P B ) )  +  ( 1  x.  ( A P B ) ) ) )
35 adddir 9396 . . . . . . . . 9  |-  ( ( k  e.  CC  /\  1  e.  CC  /\  ( A P B )  e.  CC )  ->  (
( k  +  1 )  x.  ( A P B ) )  =  ( ( k  x.  ( A P B ) )  +  ( 1  x.  ( A P B ) ) ) )
362, 35mp3an2 1302 . . . . . . . 8  |-  ( ( k  e.  CC  /\  ( A P B )  e.  CC )  -> 
( ( k  +  1 )  x.  ( A P B ) )  =  ( ( k  x.  ( A P B ) )  +  ( 1  x.  ( A P B ) ) ) )
371, 21, 36syl2an 477 . . . . . . 7  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( k  +  1 )  x.  ( A P B ) )  =  ( ( k  x.  ( A P B ) )  +  ( 1  x.  ( A P B ) ) ) )
3837adantr 465 . . . . . 6  |-  ( ( ( k  e.  NN0  /\  A  e.  X )  /\  ( ( k S A ) P B )  =  ( k  x.  ( A P B ) ) )  ->  ( (
k  +  1 )  x.  ( A P B ) )  =  ( ( k  x.  ( A P B ) )  +  ( 1  x.  ( A P B ) ) ) )
3934, 38eqtr4d 2478 . . . . 5  |-  ( ( ( k  e.  NN0  /\  A  e.  X )  /\  ( ( k S A ) P B )  =  ( k  x.  ( A P B ) ) )  ->  ( (
( k  +  1 ) S A ) P B )  =  ( ( k  +  1 )  x.  ( A P B ) ) )
4039exp31 604 . . . 4  |-  ( k  e.  NN0  ->  ( A  e.  X  ->  (
( ( k S A ) P B )  =  ( k  x.  ( A P B ) )  -> 
( ( ( k  +  1 ) S A ) P B )  =  ( ( k  +  1 )  x.  ( A P B ) ) ) ) )
4140a2d 26 . . 3  |-  ( k  e.  NN0  ->  ( ( A  e.  X  -> 
( ( k S A ) P B )  =  ( k  x.  ( A P B ) ) )  ->  ( A  e.  X  ->  ( (
( k  +  1 ) S A ) P B )  =  ( ( k  +  1 )  x.  ( A P B ) ) ) ) )
42 eqid 2443 . . . . . 6  |-  ( 0vec `  U )  =  (
0vec `  U )
435, 42, 19dip0l 24135 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
( 0vec `  U ) P B )  =  0 )
444, 18, 43mp2an 672 . . . 4  |-  ( (
0vec `  U ) P B )  =  0
455, 7, 42nv0 24036 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
0 S A )  =  ( 0vec `  U
) )
464, 45mpan 670 . . . . 5  |-  ( A  e.  X  ->  (
0 S A )  =  ( 0vec `  U
) )
4746oveq1d 6125 . . . 4  |-  ( A  e.  X  ->  (
( 0 S A ) P B )  =  ( ( 0vec `  U ) P B ) )
4821mul02d 9586 . . . 4  |-  ( A  e.  X  ->  (
0  x.  ( A P B ) )  =  0 )
4944, 47, 483eqtr4a 2501 . . 3  |-  ( A  e.  X  ->  (
( 0 S A ) P B )  =  ( 0  x.  ( A P B ) ) )
50 oveq1 6117 . . . . . 6  |-  ( j  =  0  ->  (
j S A )  =  ( 0 S A ) )
5150oveq1d 6125 . . . . 5  |-  ( j  =  0  ->  (
( j S A ) P B )  =  ( ( 0 S A ) P B ) )
52 oveq1 6117 . . . . 5  |-  ( j  =  0  ->  (
j  x.  ( A P B ) )  =  ( 0  x.  ( A P B ) ) )
5351, 52eqeq12d 2457 . . . 4  |-  ( j  =  0  ->  (
( ( j S A ) P B )  =  ( j  x.  ( A P B ) )  <->  ( (
0 S A ) P B )  =  ( 0  x.  ( A P B ) ) ) )
5453imbi2d 316 . . 3  |-  ( j  =  0  ->  (
( A  e.  X  ->  ( ( j S A ) P B )  =  ( j  x.  ( A P B ) ) )  <-> 
( A  e.  X  ->  ( ( 0 S A ) P B )  =  ( 0  x.  ( A P B ) ) ) ) )
55 oveq1 6117 . . . . . 6  |-  ( j  =  k  ->  (
j S A )  =  ( k S A ) )
5655oveq1d 6125 . . . . 5  |-  ( j  =  k  ->  (
( j S A ) P B )  =  ( ( k S A ) P B ) )
57 oveq1 6117 . . . . 5  |-  ( j  =  k  ->  (
j  x.  ( A P B ) )  =  ( k  x.  ( A P B ) ) )
5856, 57eqeq12d 2457 . . . 4  |-  ( j  =  k  ->  (
( ( j S A ) P B )  =  ( j  x.  ( A P B ) )  <->  ( (
k S A ) P B )  =  ( k  x.  ( A P B ) ) ) )
5958imbi2d 316 . . 3  |-  ( j  =  k  ->  (
( A  e.  X  ->  ( ( j S A ) P B )  =  ( j  x.  ( A P B ) ) )  <-> 
( A  e.  X  ->  ( ( k S A ) P B )  =  ( k  x.  ( A P B ) ) ) ) )
60 oveq1 6117 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  (
j S A )  =  ( ( k  +  1 ) S A ) )
6160oveq1d 6125 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
( j S A ) P B )  =  ( ( ( k  +  1 ) S A ) P B ) )
62 oveq1 6117 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
j  x.  ( A P B ) )  =  ( ( k  +  1 )  x.  ( A P B ) ) )
6361, 62eqeq12d 2457 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( ( j S A ) P B )  =  ( j  x.  ( A P B ) )  <->  ( (
( k  +  1 ) S A ) P B )  =  ( ( k  +  1 )  x.  ( A P B ) ) ) )
6463imbi2d 316 . . 3  |-  ( j  =  ( k  +  1 )  ->  (
( A  e.  X  ->  ( ( j S A ) P B )  =  ( j  x.  ( A P B ) ) )  <-> 
( A  e.  X  ->  ( ( ( k  +  1 ) S A ) P B )  =  ( ( k  +  1 )  x.  ( A P B ) ) ) ) )
65 oveq1 6117 . . . . . 6  |-  ( j  =  N  ->  (
j S A )  =  ( N S A ) )
6665oveq1d 6125 . . . . 5  |-  ( j  =  N  ->  (
( j S A ) P B )  =  ( ( N S A ) P B ) )
67 oveq1 6117 . . . . 5  |-  ( j  =  N  ->  (
j  x.  ( A P B ) )  =  ( N  x.  ( A P B ) ) )
6866, 67eqeq12d 2457 . . . 4  |-  ( j  =  N  ->  (
( ( j S A ) P B )  =  ( j  x.  ( A P B ) )  <->  ( ( N S A ) P B )  =  ( N  x.  ( A P B ) ) ) )
6968imbi2d 316 . . 3  |-  ( j  =  N  ->  (
( A  e.  X  ->  ( ( j S A ) P B )  =  ( j  x.  ( A P B ) ) )  <-> 
( A  e.  X  ->  ( ( N S A ) P B )  =  ( N  x.  ( A P B ) ) ) ) )
7041, 49, 54, 59, 64, 69nn0indALT 10758 . 2  |-  ( N  e.  NN0  ->  ( A  e.  X  ->  (
( N S A ) P B )  =  ( N  x.  ( A P B ) ) ) )
7170imp 429 1  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( ( N S A ) P B )  =  ( N  x.  ( A P B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   ` cfv 5437  (class class class)co 6110   CCcc 9299   0cc0 9301   1c1 9302    + caddc 9304    x. cmul 9306   NN0cn0 10598   NrmCVeccnv 23981   +vcpv 23982   BaseSetcba 23983   .sOLDcns 23984   0veccn0v 23985   .iOLDcdip 24114   CPreHil OLDccphlo 24231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-inf2 7866  ax-cnex 9357  ax-resscn 9358  ax-1cn 9359  ax-icn 9360  ax-addcl 9361  ax-addrcl 9362  ax-mulcl 9363  ax-mulrcl 9364  ax-mulcom 9365  ax-addass 9366  ax-mulass 9367  ax-distr 9368  ax-i2m1 9369  ax-1ne0 9370  ax-1rid 9371  ax-rnegex 9372  ax-rrecex 9373  ax-cnre 9374  ax-pre-lttri 9375  ax-pre-lttrn 9376  ax-pre-ltadd 9377  ax-pre-mulgt0 9378  ax-pre-sup 9379
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2739  df-rex 2740  df-reu 2741  df-rmo 2742  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-int 4148  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-se 4699  df-we 4700  df-ord 4741  df-on 4742  df-lim 4743  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-isom 5446  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-om 6496  df-1st 6596  df-2nd 6597  df-recs 6851  df-rdg 6885  df-1o 6939  df-oadd 6943  df-er 7120  df-en 7330  df-dom 7331  df-sdom 7332  df-fin 7333  df-sup 7710  df-oi 7743  df-card 8128  df-pnf 9439  df-mnf 9440  df-xr 9441  df-ltxr 9442  df-le 9443  df-sub 9616  df-neg 9617  df-div 10013  df-nn 10342  df-2 10399  df-3 10400  df-4 10401  df-n0 10599  df-z 10666  df-uz 10881  df-rp 11011  df-fz 11457  df-fzo 11568  df-seq 11826  df-exp 11885  df-hash 12123  df-cj 12607  df-re 12608  df-im 12609  df-sqr 12743  df-abs 12744  df-clim 12985  df-sum 13183  df-grpo 23697  df-gid 23698  df-ginv 23699  df-ablo 23788  df-vc 23943  df-nv 23989  df-va 23992  df-ba 23993  df-sm 23994  df-0v 23995  df-nmcv 23997  df-dip 24115  df-ph 24232
This theorem is referenced by:  ipasslem2  24251  ipasslem3  24252  ipasslem4  24253
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